Image Processing Reference
In-Depth Information
being a edge-like metric Gabriel center, being on the way back from a marked cell
and its closest seed, and being between two marked cells in the neighborhood:
⊧
⊨
⊩
x
∈
P
;
{
z
∈
N
(
x
)
|
dist
t
(
z
)=
dist
t
+
1
(
x
)
−
1
}
diam. 2 ;
mark
t
+
1
(
x
)=
∃
y
∈
N
(
x
)
;
{
z
∈
N
(
xy
)
|
dist
t
(
z
)=
dist
t
(
xy
)
−
1
.
5
}
diam. 3 ;
∃
y
∈
N
(
x
)
;dist
t
(
y
)=
dist
t
(
x
)+
1mod3
∧
mark
t
(
y
)
;
∃
y
0
,
y
1
∈
N
(
x
)
;mark
t
(
y
0
)
∧
mark
t
(
y
1
)
∧
x
∈
[
y
0
,
y
1
]
.
This construction therefore requires 7 states: 3 states for the distances modulo 3,
multiplied by 2 states for the mark field, plus 1 special state for the seeds that always
have dist
. Its neighborhood radius is 2 because of the edge-like
metric Gabriel centers detection. Secondly, if one wants to consider moving seeds
for example, it is necessary to keep the distinction between the fields cent, back and
conv.
=
0andmark
=
Acknowledgements.
The authors would like to thank A. Adamatzky for his important in-
volvement in the development of this work. He was the one who incited them to begin this
work and also the one who told them about Gabriel graphs when they were seeking for
the appropriate subgraphs of Delaunay graphs which is naturally computed by the pairwise
algorithm.
References
1. Adamatzky, A.: Voronoi-like partition of lattice in cellular automata. Mathematical and
Computer Modelling 23, 51-66 (1996), doi:doi:10.1016/0895-7177(96)00003-9
2. Aupetit, M., Catz, T.: High-dimensional labeled data analysis with topology representing
graphs. Neurocomputing 63, 139-169 (2005)
3. Aurenhammer, F.: Voronoi diagrams - a survey of a fundamental geometric data struc-
ture. ACM Comput. Surv. 23(3), 345-405 (1991),
4. Chen, H., Wei, W.: Geodesic Gabriel graph based supervised nonlinear manifold learn-
ing. In: Huang, D.-S., Li, K., Irwin, G.W. (eds.) ICIC 2006. LNCIS, vol. 345, pp. 882-
887. Springer, Heidelberg (2006)
5. Clarridge, A.G., Salomaa, K.: An improved cellular automata based algorithm for the
45-convex hull problem. Journal of Cellular Automata 5(1-2), 107-120 (2010)
6. Gabriel, R.K., Sokal, R.R.: A new statistical approach to geographic variation analysis.
Systematic Zoology 18(3), 259-278 (1969)
7. Ilachinski, A.: Cellular Automata: A Discrete Universe. World Scientific Publishing Co.,
Inc., River Edge (2001)
8. Kanj, I.A., Perkovic, L., Xia, G.: Local construction of near-optimal power spanners
for wireless ad hoc networks. IEEE Transactions on Mobile Computing 8(4), 460-474
(2009),
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